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Degenerate Cahn-Hilliard and incompressible limit of a Keller-Segel model

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arXiv 2021年

作者： Elbar, Charles Perthame, Benoît Poulain, Alexandre ParisF-75005 France Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn-Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy’s law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller-Segel system to the Cahn-Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller-Segel system and the relaxed degenerate Cahn-Hilliard *** Codes 35B40, 35B45, 35G20, 35Q92 Copyright © 2021, The Authors. All rights reserved.

关键词： Asymptotic analysis

ON THE IMPROVED CONVERGENCE OF LIFTED DISTRIBUTIONAL GAUSS CURVATURE FROM REGGE ELEMENTS

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arXiv 2024年

作者： Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max Portland State University PO Box 751 PortlandOR97201 United States Institute of Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 Wien1040 Austria Institute of Numerical and Applied Mathematics University of Göttingen Lotzestr. 16-18 Göttingen37083 Germany

Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical *** Codes 65N30 (Primary) 53A70, 83C27 (Secondary) Copyright © 2024, The Authors. All rights reserved.

关键词： Tensors

ROBUST APPROXIMATION of GENERALIZED BIOT-BRINKMAN PROBLEMS

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arXiv 2021年

作者： Hong, Qingguo Kraus, Johannes Kuchta, Miroslav Lymbery, Maria Mardal, Kent-André Rognes, Marie E. Department of Mathematics Pennsylvania State University United States Faculty of Mathematics University of Duisburg-Essen Germany Department of Numerical Analysis and Scientific Computing Simula Research Labaratory Faculty of Mathematics University of Duisburg-Essen Germany Department of Mathematics University of Oslo Norway Department for Numerical Analysis and Scientific Computing Simula Research Laboratory Department of Numerical Analysis and Scientific Computing Simula Research Labaratory Department of Mathematics University of Bergen Norway

The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical *** Codes 65F08, 65F10, 65M12, 65M22, 65M60, 65N22, 65N30, 92C05, 92C10 Copyright © 2021, The Authors. All rights reserved.

关键词： Finite element method

A Risk Management Perspective on Statistical Estimation and Generalized Variational Inference

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arXiv 2023年

作者： Javeed, Aurya S. Kouri, Drew P. Surowiec, Thomas M. Optimization and Uncertainty Quantification Sandia National Laboratories P.O. Box 5800 MS-1320 AlbuquerqueNM87125 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway

Generalized variational inference (GVI) provides an optimization-theoretic framework for statistical estimation that encapsulates many traditional estimation procedures. The typical GVI problem is to compute a distribution of parameters that maximizes the expected payoff minus the divergence of the distribution from a specified prior. In this way, GVI enables likelihood-free estimation with the ability to control the influence of the prior by tuning the so-called learning rate. Recently, GVI was shown to outperform traditional Bayesian inference when the model and prior distribution are misspecified. In this paper, we introduce and analyze a new GVI formulation based on utility theory and risk management. Our formulation is to maximize the expected payoff while enforcing constraints on the maximizing distribution. We recover the original GVI distribution by choosing the feasible set to include a constraint on the divergence of the distribution from the prior. In doing so, we automatically determine the learning rate as the Lagrange multiplier for the constraint. In this setting, we are able to transform the infinite-dimensional estimation problem into a two-dimensional convex program. This reformulation further provides an analytic expression for the optimal density of parameters. In addition, we prove asymptotic consistency results for empirical approximations of our optimal distributions. Throughout, we draw connections between our estimation procedure and risk management. In fact, we demonstrate that our estimation procedure is equivalent to evaluating a risk measure. We test our procedure on an estimation problem with a misspecified model and prior distribution, and conclude with some extensions of our *** Codes 62A10, 62A15, 62C12, 62G05, 62G07 © 2023, CC BY.

关键词： Risk management

A PARALLEL RANK-ADAPTIVE INTEGRATOR FOR DYNAMICAL LOW-RANK APPROXIMATION

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arXiv 2023年

作者： Ceruti, Gianluca Kusch, Jonas Lubich, Christian Institute of Mathematics EPF Lausanne Lausanne1015 Switzerland Numerical Analysis and Scientific Computing Universität Innsbruck Technikerstraße 13 Innsbruck6020 Austria Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 Tübingen72076 Germany

This work introduces a parallel and rank-adaptive matrix integrator for dynamical low-rank approximation. The method is related to the previously proposed rank-adaptive basis update & Galerkin (BUG) integrator but differs significantly in that all arising differential equations, both for the basis and the Galerkin coefficients, are solved in parallel. Moreover, this approach eliminates the need for a potentially costly coefficient update with augmented basis matrices. The integrator also incorporates a new step rejection strategy that enhances the robustness of both the parallel integrator and the BUG integrator. By construction, the parallel integrator inherits the robust error bound of the BUG and projector-splitting integrators. Comparisons of the parallel and BUG integrators are presented by a series of numerical experiments which demonstrate the efficiency of the proposed method, for problems from radiative transfer and radiation therapy. Copyright © 2023, The Authors. All rights reserved.

关键词： Differential equations

Spatial modeling algorithms for reactions and transport (SMART) in biological cells

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arXiv 2024年

作者： Francis, Emmet A. Laughlin, Justin G. Dokken, Jørgen S. Finsberg, Henrik N.T. Lee, Christopher T. Rognes, Marie E. Rangamani, Padmini Department of Mechanical and Aerospace Engineering University of California La Jolla San DiegoCA United States Computational Engineering Division Lawrence Livermore National Laboratory LivermoreCA United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Computational Physiology Simula Research Laboratory Oslo Norway

Biological cells rely on precise spatiotemporal coordination of biochemical reactions to control their many functions. Such cell signaling networks have been a common focus for mathematical models, but they remain challenging to simulate, particularly in realistic cell geometries. Herein, we present our software, Spatial Modeling Algorithms for Reaction and Transport (SMART), a package that takes in high-level user specifications about cell signaling networks and molecular transport, and then assembles and solves the associated mathematical and computational systems. SMART uses state-of-the-art finite element analysis, via the FEniCS Project software, to efficiently and accurately resolve cell signaling events over discretized cellular and subcellular geometries. We demonstrate its application to several different biological systems, including YAP/TAZ mechanotransduction, calcium signaling in neurons and cardiomyocytes, and ATP generation in mitochondria. Throughout, we utilize experimentally-derived realistic cellular geometries represented by well-conditioned tetrahedral meshes. These scenarios demonstrate the applicability, flexibility, accuracy and efficiency of SMART across a range of temporal and spatial scales. © 2024, CC BY.

关键词： Cell signaling

Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces

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arXiv 2022年

作者： Milz, Johannes Surowiec, Thomas M. H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology AtlantaGA30332 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the *** Codes 90C15, 90C06, 62F12, 35Q93, 49M41, 49J52 © 2022, CC BY-SA.

关键词： Stochastic programming

Deep neural networks on diffeomorphism groups for optimal shape reparameterization

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arXiv 2022年

作者： Celledoni, Elena Glöckner, Helge Riseth, Jørgen Schmeding, Alexander Department of Mathematics NTNU Trondheim Norway Institut für Mathematik Universität Paderborn Germany Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Norway Department of Mathematics University of Oslo Norway

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting *** Codes 65K10, 58D05, 46T10 © 2022, CC BY.

关键词： Deep neural networks

Natural convection in the horizontal annulus: critical Rayleigh number for the steady problem

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arXiv 2025年

作者： Passerini, Arianna Rummler, Bernd Růžička, Michael Thäter, Gudrun Department of Mathematics and Scientific Computing Università di Ferrara Via Saragat 1 Ferrara44100 Italy Institute for Analysis and Numerics Otto-von-Guericke-Universität Magdeburg PF 4120 Magdeburg39016 Germany Institute for Applied Mathematics Universität Freiburg Eckerstr. 1 Freiburg79104 Germany Institute for Applied & Numerical Mathematics KIT Karlsruhe76128 Germany

For the 2D Oberbeck-Boussinesq system in an annulus we are looking for the critical Rayleigh number for which the (nonzero) basic flow loses stability. For this we consider the corresponding Euler-Lagrange equations and construct a precise functional analytical frame for the Laplace- and the Stokes problem as well as the Bilaplacian operator in this domain. With this frame and the right set of basis functions it is then possible to construct and apply a numerical scheme providing the critical Rayleigh number. These results will be published in ZAMM in *** Codes 76E09, 76E06, 76D03, 35Q35 © 2025, CC BY.

关键词： Rayleigh number